Classification of some 3-subgroups of the finite groups of Lie type E6

Jianbei An, Heiko Dietrich, Shih-Chang Huang

Research output: Contribution to journalArticle

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Abstract

We consider the finite exceptional group of Lie type G=E6 ε(q) (universal version) with 3|q−ε, where E6 +1(q)=E6(q) and E6 −1(q)=2E6(q). We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local M<G which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing Z(G), all extraspecial subgroups containing Z(G), and all cyclic groups of order 9 containing Z(G). These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.

Original languageEnglish
Pages (from-to)4020-4039
Number of pages20
JournalJournal of Pure and Applied Algebra
Volume222
Issue number12
DOIs
Publication statusPublished - 2018 Dec 1

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Finite Groups of Lie Type
Subgroup
Conjugacy
Groups of Lie Type
Modular Representations
Cyclic group
Representation Theory
Inclusion
Classify

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

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title = "Classification of some 3-subgroups of the finite groups of Lie type E6",
abstract = "We consider the finite exceptional group of Lie type G=E6 ε(q) (universal version) with 3|q−ε, where E6 +1(q)=E6(q) and E6 −1(q)=2E6(q). We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local M<G which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing Z(G), all extraspecial subgroups containing Z(G), and all cyclic groups of order 9 containing Z(G). These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.",
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Classification of some 3-subgroups of the finite groups of Lie type E6 . / An, Jianbei; Dietrich, Heiko; Huang, Shih-Chang.

In: Journal of Pure and Applied Algebra, Vol. 222, No. 12, 01.12.2018, p. 4020-4039.

Research output: Contribution to journalArticle

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